SOLUTION: A farmer plans to build a rectangular pen taht is 16 feet long. The area of the pen is 80 square feet. Is 40 feet of fencing enough to fence in the pen?

Let the length be x
Let the width by y

Area = xy = 80
Perimeter = 2x + 2y < 40 

So we have 



Solve the equation for y 

Substitute in the inequality





Since x is positive we can multiply through by x without
reversing the inequality symbol





Divide through by 2



Find the zeros of the expression on the left.

It doesn't factor, so:

 

 

 

 









The two zeros are approximately 5.5 and 14.5

So we mark those two points on a number line:




So we test a point is the region left of 5.5, say 5, in

the region left of the 5 in the inequality:






That's false so we do not shade the part of the number line
left of the green point.

Next we test a point is the region between 5.5 and 14.5, say 6, in
the inequality:






That's true so we shade the part of the number line between the
green and red points:



Now we test a point is the region right of 14.5, say 15, in

the region left of the 5 in the inequality:






That's false so we do not shade the part of the number line
right of the red point.

So the solution set for the inequality in interval notation is



So yes we can make the length any value  of x between those two
values, and the width , and 40 will be enough fencing.

Edwin